If you start with the story mode this will likely be one of your first achievements after Cast Member:
Of course if you jump into Arena mode it's entirely possible to earn enough gems to get this achievement first, but it really doesn't matter. In Arena mode you earn gems based on your performance. The better you do the more gems you get.
In story mode you earn gems with each level completed. 3 are required for the exit to unlock, but you will likely earn many more. Most levels have 7 gems but some only have 6. Also of note is if you get all the gems, the yarn, and get the time bonus you get 2 extra gems. Go you!
The starting cost for prisoners is 10 gems each (it goes up over time) so the absolute minimum level completion is 2 and if you only get 3 gems in each level you will have the necessary number of gems to purchase one prisoner after 4 levels.
Next go to the gift shop found on the lower level of each stage and make your way to any of the prisoner purchase devices (which I have since learned are called Chonku machines). Buy a prisoner and POP achievement unlocked. You will want to keep spending gems to unlock more prisoners to get the following achievements: